cp's OEIS Frontend

A339405 Number of partitions of n into an odd number of parts that are not multiples of 3.

%I A339405 #6 Dec 03 2020 12:20:07
%S A339405 0,1,1,1,2,3,3,5,6,8,11,14,17,23,28,35,44,55,66,83,100,122,148,179,
%T A339405 213,259,307,366,436,518,609,723,848,997,1169,1369,1593,1864,2163,
%U A339405 2513,2914,3376,3894,4503,5182,5965,6854,7869,9008,10325,11794,13470,15363,17509,19911,22654,25713,29177
%N A339405 Number of partitions of n into an odd number of parts that are not multiples of 3.
%H A339405 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A339405 G.f.: (1/2) * (Product_{k>=1} (1 - x^(3*k)) / (1 - x^k) - Product_{k>=1} (1 + x^(3*k)) / (1 + x^k)).
%F A339405 a(n) = (A000726(n) - A109389(n)) / 2.
%e A339405 a(7) = 5 because we have [7], [5, 1, 1], [4, 2, 1], [2, 2, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1].
%p A339405 b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
%p A339405       b(n, i-1, t)+`if`(irem(i, 3)=0, 0, b(n-i, min(n-i, i), 1-t))))
%p A339405     end:
%p A339405 a:= n-> b(n$2, 0):
%p A339405 seq(a(n), n=0..60);  # _Alois P. Heinz_, Dec 03 2020
%t A339405 nmax = 57; CoefficientList[Series[(1/2) (Product[(1 - x^(3 k))/(1 - x^k), {k, 1, nmax}] - Product[(1 + x^(3 k))/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
%Y A339405 Cf. A000726, A001651, A027193, A109389, A339404, A339406, A339407.
%K A339405 nonn
%O A339405 0,5
%A A339405 _Ilya Gutkovskiy_, Dec 03 2020